Mechanics of Machines Notes

ME 010601 Mechanics of Machines

Module - 2

Governors

Binil BabuDepartment of Mechanical Engineering

SNMIMT,Maliankara

Contents

1 Overview 1

2 Terminologies 1

3 Classification of governors 3

4 Watt governor 4

5 Porter governor 4

6 Proell Governor 6

7 Hartnell governor 7

8 Effort and power of a governor 9

9 Sensitiveness of a governor 129.1 Stability of governor . . . . . . . . . . . . . . . . . . . . . . . 129.2 Isochronous governor . . . . . . . . . . . . . . . . . . . . . . . 129.3 Hunting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

10 Inertia governor 13

11 Previous year university questions 14

12 Reference 15

i

Binil Babu,ME Dept. 1

GOVERNORS

1 Overview

A governor, or speed limiter, is a device used to regulate the speed of amachine, such as an engine. Centrifugal governor,also known as watt gover-nor invented by James Watt in 17th century to regulate the speed of steamengine by altering steam flow. The main objective of this chapter is to un-derstand the basic terminologies,classification, working of governors and itsapplication.The governors are mainly classified into centrifugal and Inertiagovernor.The function of a governor is to maintain the speed of an enginewithin specified limits whenever there is variation in load.When the speed ofengine varies in each revolution,(cyclic variation),it is due to variation inoutput torque of engine.It can be regulated by mounting a suitable flywheel onthe shaft.The working of flywheel is continuous and intermittent in the caseof governor.

2 Terminologies

Height of governor is the vertical distance from the plane of rotation ofthe balls to the point of intersection of the upper arms along the axis of thespindle. It is usually denoted by ”h”. The height of governor decreases withincrease in speed and vise versa.

Equilibrium speed :It is the speed at which governor balls,arms etc.arein complete equilibrium and the sleeve does not tend to move upwards ordownwards.

Mean equilibrium speed:It is the speed at the mean position of theballs or the sleeve.

Maximum and minimum equilibrium speeds:The speeds at themaximum and minimum radius of rotation of the balls,without tending tomove either way are known as maximum and minimum equilibrium speedsrespectively.

Sleeve lift:It is the vertical distance which the sleeve travels due tochange in equilibrium speed.


Radius of rotation:It is the horizontal distance from the axis to thecentre of to centre of flyball.

Insensitiveness of governor:The friction force at the sleeve gives in-crease to the insensitiveness in the governor. At any specified radius thereshall be two different speeds one being while sleeve moves up and other whilesleeve moves down. Given figure illustrates the controlling force diagram forsuch as governor.

The equivalent three values of speeds for the similar radius OA are fol-lowing:i.The speed N while there is no friction.ii.The speed N

′while speed is rising or sleeve is on the verge of moving

up,andiii.The speed N

′′while speed is dropping or sleeve on the verge of moving

down.

It means that, while radius is OA, the speed of rotation might varyamong the restrict N

′′and N

′, without any displacement of the governor

sleeve. The governor is call to be insensitiveness over this range of speed.So,Coefficient of insensitiveness = N ′′−N ′

N


3 Classification of governors

The governors are mainly classified into (i) Centrifugal governor(ii)InertiagovernorCentrifugal governor is again classified into :-

1. Gravity controlled centrifugal governor : In this type of governorgravity force to weight on sleeve or weight of sleeve itself control themovement of sleeve.

2. Spring controlled centrifugal governor : In this governors helicalspring is used to control the movement of sleeve or balls.

Figure 1: classification of centrifugal governor


4 Watt governor

Figure 2: Watt Governor

It is the simplest form of governor.The main parts of this governor issleeve,spindle,flyballs etc..The action of this governor depends upon the cen-trifugal effects produced by the masses of two balls.

5 Porter governor

Figure 3: Porter governor

The only difference between watt and porter governor is the mass addedon the sleeve of governor.The mass added on the sleeve helps the governor


to retain its equilibrium position as fast as possible.

m = mass of ball in kgw = weight of ball in NM = mass of central load in kgW = weight of central load in Nr = radius of rotation in metresh = height of governor in metresN = speed of balls in rpmω = Angular speed of ball in radFC = Centrifugal forceT1 = Force in arms in NT2 = Force in the links in Nα = Angle of inclination of arm to vertical axisβ = angle of inclination of link to vertical axis

N2 = 895h× mg+Mg±F

2(1+q)

mg, is the equation to find speed when friction is

considered. Here q = tanαtanβ

when α = β and q = tanβtanα

when α 6= β

N2 = 895h× [m+M

m], is the equation to find speed when there is no friction on

the sleeve. In this case α = β


6 Proell Governor

Figure 4: Proell Governor

N2 = FMBM× [

m+M2(1+q)

m]× 895

h, when α 6= β

N2 = FMBM× [m+M

m]× 895

h, when α = β

m = mass of fly ballM = mass on sleeveq = tanβ

tanβ

α = Angle of inclination of arm to vertical axisβ = angle of inclination of link to vertical axisIf we consider the equation for equilibrium speed of porter governor and proellgovernor we can identify that the equilibrium speed reduces for given valuesof m, M and h. That means we can use smaller masses in proell governorthan porter governor.


7 Hartnell governor

Figure 5: Hartnell Governor

It is a Spring loaded Governor . It has two bell crank levers carryingfly ball at one end and roller attached to other end, the function of springis provide the counter force which acts against centrifugal force. The springand shaft is enclosed inside a casing .The sleeve is pressed against the springwhen the centrifugal force on the balls increases. Due to spring return naturethis governor can be mounded in horizontal,inverted, (inclined) position.A Hartnell governor is a spring loaded governor as shown in Fig.5. It con-sists of two bell crank levers pivoted at the points O,O to the frame. Theframe is attached to the governor spindle and therefore rotates with it. Eachlever carries a ball at the end of the vertical arm OB and a roller at the endof the horizontal arm OR. A helical spring in compression pro- vides equaldownward forces on the two rollers through a collar on the sleeve. The springforce may be adjusted by screwing a nut up or down on the sleeve.

m = Mass of each ball in kg,

M = Mass of sleeve in kg,

r1 = Minimum radius of rotation in metres

r2 = Maximum radius of rotation in metres


ω1 = Angular speed of the governor at minimum radius in rad/s,

ω2 = Angular speed of the governor at maximum radius in rad/s,

S1 = Spring force exerted on the sleeve at ω1 in newtons,

S2 = Spring force exerted on the sleeve at ω2 in newtons,

FC1 = Centrifugal force at ω1 in newtons = mω21r1,

FC2 = Centrifugal force at ω2 in newtons = mω22r2

s = Stiffness of the spring or the force required to compress the springby one mm,

x = Length of the vertical or ball arm of the lever in metres,

y = Length of the horizontal or sleeve arm of the lever in metres,

r = Distance of fulcrum O from the governor axis or the radius of rota-tion when the governor is in mid-position, in metres.

Equations required to solve problems

h = (r2 − r1)× yx

S2–S1 = h.s,

s = S2−S1

h= S2−S1

r2−r1 ×xy

Neglecting obliquity effect, moment due to weight at minimum position,M g +S1 = 2FC1 × x

y, similarly for maximum position M g +S2 = 2FC2 × x

y

S2 − S1 = 2(FC2 − FC1)× xy

, substitute S2 − S1 = h.s, s = S2−S1

r2−r1 × [xy]2

FC = FC1 + (FC2 − FC1)r−r1r2−r1 = FC2 − (FC2 − FC1)

r2−rr2−r1

• We can neglect obliquity effect unless it is mentioned in the question


• FC is the centrifugal force for any intermediate position and r is itscorresponding radius of rotation.

8 Effort and power of a governor

Governor effort and power can be used to compare the effectiveness of dif-ferent type of governors.

Governor Effort:It is defined as the mean force exerted on the sleeveduring a given change in speed. When governor speed is constant the netforce at the sleeve is zero. When governor speed increases, there will be a netforce on the sleeve to move it upwards and sleeve starts moving to the newequilibrium position where net force becomes zero.

Governor Power: It is defined as the work done at the sleeve for agiven change in speed. Therefore, Power of governor = Governor effort× Displacement of sleeve

N = Equilibrium speed corresponding to configuration (a)c = Increased percentage in speed, increase in speed is c.NIncreased speed = N+c.N = N(1+c).

Figure 6: Effort and power of watt governor

When the speed is N rpm the sleeve load is Mg and we also assumingthat the angle α = β

Let height of the governor h for speed given byh = m+M

m× 895

N2 ...............(1)when the speed of governor increased to (1 + c)2N2 to maintain the height ofgovernor ”h” we increase the mass on the sleeve ie; M to M1 now,the heightof governor,h = m+M1

2× 895

(1+c)2N2 .................(2)


equate equation (1) and (2)

m+M =m+M1

(1+c)2

m+M × (1 + c)2 = m+M1

M1 = (m+M)× (1 + c)2 −m

M1 −M = (m+M)× (1 + c)2 −m−M

M1 −M = (m+M)[(1 + c)2 − 1] ...........(3)

(M1 − M)g is the downward force which must be applied in order toprevent the sleeve from rising as speed increases.When the sleeve rises andreaches new position this downward force gradually diminishes to zero.Let”P” be the mean force exerted on the sleeve during increase in speed or theeffort of the governor

P = (M1−M)g2

= (m+M)[(1+c)2−1]g2

P = (m+M)[1+2c+c2−1]g2

P = c(m+M)g.........(4)

If we consider frictional force ”F”, then effortP = c(mg+Mg± F)

Power of governor is given by, Power = Mean effort × lift ofsleeve (x)........(5)

Power = P × x

Sleeve lift,x

Let height of governor at speed N is ”h” and at the speed


(1 + c)2 N is ”h1”, then the lift of sleeve x can be written asx = (h− h1)

h = 895N2 × m+M

m at speed is Nrpm

h1 = 895(1+c)2N2 × m+M

m at speed is (1 + c)2N 2rpm

h1/h,

h1

h = 1(1+c)2

h1 = h(1+c)2 , x = 2(h1 − h) = 2[h− h

(1+c)2 ] = 2h[1− 1(1+c)2 ]

x = 2h[1+c2+2c−11+c2+2c ]

x = 2h[ 2c(1+2c) ]............(6)

Now substitute value of x and P in equation of Power ofgovernor

Governor power = c(m+M)g×2h[ 2c1=2c ] , (the value of c2 can

be neglected since it is very small)

Power = 4c2

1+2c(m+M)gh......(7)

The effort of a governor is the mean force exerted atthe sleeve for a given percentage change of speed or lift of thesleeve.The power of a governor is the work done at the sleeve for agiven percentage of change in speed. It is the product of meaneffort and the distance through which the sleeve moves.


9 Sensitiveness of a governor

Sensitiveness is the the ratio of difference between maximumand minimum equilibrium speed to the mean equilibrium speed.N1= Min. equilibrium speedN2 = Max. equilibrium speedN = Mean speed = N1+N2

2

Therefore, Sensitiveness of governor = N1−N2

N = 2(N1−N2)N1+N2

9.1 Stability of governor

A governor is said to be stable when for every speed within theworking range there is a definite configuration, ie; there is onlyone radius of rotation for the governor in equilibrium condition.

9.2 Isochronous governor

A governor is said to be isochronous when the equilibriumspeed is constant.(ie; the range of speed is zero)for all radii ofrotation of the balls within the working range,neglecting fric-tion.Isochronism is the stage of infinite sensitivity.A porter gov-ernor cannot be isochronous.

9.3 Hunting

A governor is said to be hunt if the speed of the engine fluctuatescontinuously above and below the mean speed.This is caused bytoo sensitive governor which changes the fuel supply by a largeamount when a small change in speed of rotation take place.


10 Inertia governor

Figure 7: Inertia governor

In inertia governors, the balls are arranged in manner thatthe inertia forces caused by angular acceleration or retardationof the governor shaft tend to change their position. The obviousadvantage of inertia governor lies in its rapid response to theeffect of a change of load. This advantage is small, however bythe practical difficulty of arranging for the complete balance ofthe revolving parts of the governor. For this reason Centrifugalgovernors are preferred over the inertia governors.

The relative movement of governor balls is controlled by theaction of spring.The arm connecting ball is hinged at flywheelconnected to the shaft.The relative position of ball arm withrespect to the flywheel is depends on the angular velocity ω andinstantaneous angular acceleration α of the shaft.The relativemovement of the ball arm is used to control power input to anengine.


11 Previous year university questions

1. What is the fundamental difference between a gov-ernor and flywheel? (3 mark)

Flywheel which minimizes the fluctuation of speed withinthe cycle,but it cannot minimizes the fluctuations due to thevariations of load.This means flywheel does not have any con-trol over mean speed of engine.Govenor is used to minimize thefluctuations in speed due to variation in load.It has no influ-ence over cyclic fluctuation of speed,but controls mean speedfor long period during which load on engine varies.The functionof governor is to supply fuel,when load on engine increases anddecreases the fuel supply when load decreases so as to keep thespeed of engine almost constant in different loads.

2. What is insensitiveness in governor?Read page no.2

3. What are the different types of governor?Read page no.3

4.What are spring controlled governor?Describe thefunction of any one of them.

5.Explain isochronism in a governor

6.Discuss inertia governor with neat sketches.

7.Explain the function of a flywheel.


12 Reference

1. Theory of Machines by Khurmi& Gupta

2. Theory of Mechanisms and Machines by Amitabha Ghoshand Asok Kumar Mallik

3. https://www.youtube.com/watch?v=nDccEoBqc3M

4. https://www.youtube.com/watch?v=dSyYzWOSZmk